Page 80 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 65

portant and very useful to have at our disposal more general construc-

tion methods for the fluid flows. The conformal mapping will be such
a method for determining a fluid flow satisfying some “a priori” given
requirements.
Generally, a conformal transformation of a domain (d) from the plane
(z), onto a domain (D) from the plane (Z), is a holomorphic function
h,: (d) + (D) which fulfils the condition h’(z) 4 0 (the angles preserving

condition). If the conformal mapping is also univalent (injective) this
will be a conformal mapping of the domain (d) onto the domain (D).
Obviously the holomorphicity is preserved by a conformal mapping. The
same thing happens with the connection order of the domain (d). We
know that the determination of the conformal mapping (on a canonical
domain) is synonymous with that of the Green function associated to the

Laplace operator and to the involved domain, that is with the possibility
to solve a boundary value problem of Dirichlet type for the same operator
and domain [69].
Concerning the existence of conformal mapping, in the case of a
simply-connected domain, a classical result known as Riemann—Cara-
theodory’s theorem states that:

For a given simply-connected domain (d) from the plane (z) and
whose boundary contains more than a point, it is always possible to map
it conformally, in a unique manner, onto the circular disk |Z| <1 from
the plane (Z), such that to a certain point zg € (d) there corresponds an
internal given point Zo from |Z| <1 and to a certain direction passing

through zg there corresponds a given direction passing through Zg.
We remark that the uniqueness of the conformal mapping holds to
within three arbitrary parameters, so that we deal, basically, with a
class of functions which defines the considered conformal mapping.
Unfortunately the proof of the existence in this theorem is far from

being a constructive one such that, in practical problems, we are faced
with the effective determination of the conformal mapping. There are
few cases when these conformal mappings are explicitly (analytically)
found. That is why the approximative procedures (one of them being
sketched in a next section) are of the greatest interest.

Finally, the above result could also be extended to the doubly-connec-
ted domains (see, for instance, Y. Komatu [75]) and even to the general
multiply-connected domains but, in this last case, it is extremely difficult
to determine and work with the involved functions. As a consequence
the conformal mapping method is not practically used in the case of
domains with a higher order of connection.
Returning to the simply-connected case, the following result is of re-

markable interest in different applications: